Finite Group Databasebeta

← C137×#7828 #46857214

Subgroups — C137×#7828 #46857214

Order 35072.

404 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2727×C137C211yesmaximal, characteristic
2#2100×C137C211yesmaximal, characteristic, complemented
3#2100×C137C211yesmaximal, characteristic, complemented
4#2094×C137C211yesmaximal, characteristic
5#1271×C137C211yesmaximal, characteristic, complemented
6#1271×C137C211yesmaximal, characteristic, complemented
7#1270×C137C211yesmaximal, characteristic
8C2×#126×C137—21no—
9C2×#106×C137—21no—
10C2×#102×C137—21no—
11#399×C137—21no—
12#396×C137—21no—
13#394×C137—21no—
14#390×C137C2211yescharacteristic
15#389×C137—21no—
16#389×C137C2211yescharacteristic
17#389×C137C2211yescharacteristic
18#388×C137—21no—
19#384×C137C2211yescharacteristic
20#340×C137—21no—
21#340×C137—21no—
22#340×C137—21no—
23#340×C137—21no—
24#336×C137C2211yescharacteristic
25#336×C137C2211yescharacteristic
26#336×C137C2211yescharacteristic
27C22×D4×C137D411yescharacteristic
28C22×C8×C137—21no—
29C22×C8×C137—21no—
30C22×C8×C137—21no—
31#126×C137—42no—
32#126×C137—42no—
33C2×C4⋊C4×C137—21no—
34C2×C4⋊C4×C137—21no—
35C2×C4⋊C4×C137—21no—
36C2×C4⋊C4×C137—21no—
37C2×C4⋊C4×C137—21no—
38C2×C4⋊C4×C137—21no—
39C2×C4⋊C4×C137—21no—
40C2×C4⋊C4×C137D411yescharacteristic
41C2×C4⋊C4×C137D411yescharacteristic
42C2×C4⋊C4×C137D411yescharacteristic
43C2×C4⋊C4×C137D411yescharacteristic
44C2×C4⋊C4×C137D411yescharacteristic
45C2×C22⋊C4×C137—21no—
46C2×C22⋊C4×C137—21no—
47C2×C22⋊C4×C137—21no—
48C2×C22⋊C4×C137—21no—
49C2×C22⋊C4×C137—21no—
50C2×C22⋊C4×C137—21no—
51C2×C42×C137—21no—
52C2×C42×C137C2311yescharacteristic
53#106×C137—42no—
54#102×C137—42no—
55#95×C137—21no—
56#95×C137—21no—
57#95×C137—21no—
58C24×C137—21no—
59C2×D4×C137—21no—
60C2×D4×C137—21no—
61C2×D4×C137—41no—
62C2×D4×C137—41no—
63C22×C4×C137—21no—
64C22×C4×C137—21no—
65C22×C4×C137—21no—
66C22×C4×C137—21no—
67C22×C4×C137—21no—
68C22×C4×C137—41no—
69C22×C4×C137C2×D411yescharacteristic
70C22×C4×C137C2×D411yescharacteristic
71C22×C4×C137C2×D411yescharacteristic
72C2×C8×C137—22no—
73C2×C8×C137—41no—
74C2×C8×C137—41no—
75C2×C8×C137—41no—
76C2×C8×C137—41no—
77C2×C8×C137—41no—
78C2×C8×C137—41no—
79C2×C8×C137—41no—
80C2×C8×C137—41no—
81C4⋊C4×C137—21no—
82C4⋊C4×C137—21no—
83C4⋊C4×C137—21no—
84C4⋊C4×C137—41no—
85C4⋊C4×C137—41no—
86C4⋊C4×C137—41no—
87C4⋊C4×C137—42no—
88C4⋊C4×C137—41no—
89C4⋊C4×C137—41no—
90C4⋊C4×C137—42no—
91C4⋊C4×C137—42no—
92C4⋊C4×C137—41no—
93C4⋊C4×C137—41no—
94C4⋊C4×C137—41no—
95C4⋊C4×C137—41no—
96C4⋊C4×C137—81no—
97C4⋊C4×C137SD1612yes—
98C22⋊C4×C137—42no—
99C22⋊C4×C137—42no—
100C22⋊C4×C137—41no—
101C22⋊C4×C137—41no—
102C22⋊C4×C137—41no—
103C22⋊C4×C137—41no—
104C22⋊C4×C137—81no—
105C22⋊C4×C137—81no—
106C42×C137—41no—
107C42×C137—42no—
108C23×C137—41no—
109C23×C137—41no—
110C23×C137—41no—
111C23×C137—41no—
112C23×C137—41no—
113C23×C137—41no—
114C23×C137—41no—
115C23×C137#12011yescharacteristic
116D4×C137—42no—
117D4×C137—81no—
118C2×C4×C137—21no—
119C2×C4×C137—21no—
120C2×C4×C137—21no—
121C2×C4×C137—21no—
122C2×C4×C137—21no—
123C2×C4×C137—21no—
124C2×C4×C137—22no—
125C2×C4×C137—41no—
126C2×C4×C137—41no—
127C2×C4×C137—41no—
128C2×C4×C137—41no—
129C2×C4×C137—41no—
130C2×C4×C137—41no—
131C2×C4×C137—41no—
132C2×C4×C137—41no—
133C2×C4×C137—41no—
134C2×C4×C137—41no—
135C2×C4×C137—41no—
136C2×C4×C137—41no—
137C2×C4×C137—41no—
138C2×C4×C137—41no—
139C2×C4×C137—81no—
140C2×C4×C137—81no—
141C2×C4×C137—81no—
142C2×C4×C137C2×SD1611yescharacteristic
143C2×C4×C137#13511yescharacteristic
144C2×C4×C137#13511yescharacteristic
145C2×C4×C137AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C137#13711yescharacteristic
147C2×C4×C137#13711yescharacteristic
148C8×C137—42no—
149C8×C137—81no—
150C8×C137—81no—
151C22×C137—81no—
152C22×C137—81no—
153C22×C137—81no—
154C22×C137—81no—
155C22×C137—81no—
156C22×C137—81no—
157C22×C137—81no—
158C22×C137#45011yescharacteristic
159C22×C137#45211yescharacteristic
160C22×C137#45211yescharacteristic
161C22×C137#45311yescharacteristic
162C22×C137#45411yescharacteristic
163C22×C137#45411yescharacteristic
164C22×C137#45811yescharacteristic
165C4×C137—21no—
166C4×C137—21no—
167C4×C137—21no—
168C4×C137—21no—
169C4×C137—21no—
170C4×C137—21no—
171C4×C137—42no—
172C4×C137—81no—
173C4×C137—81no—
174C4×C137—81no—
175C4×C137—81no—
176C4×C137—161no—
177C2×C137—161no—
178C2×C137#161511yescharacteristic
179C2×C137#161611yescharacteristic
180C2×C137#161811yescharacteristic
181C2×C137#163111yescharacteristic
182C2×C137#163511yescharacteristic
183C2×C137#163811yescharacteristic
184C2×C137#165711yescharacteristic
185#7828C13711yesmaximal, characteristic, direct product factor, Sylow
186C137#782811yescharacteristic, direct product factor, Sylow
187#2727C2×C13711yescharacteristic
188#2100C2×C13711yescharacteristic, complemented
189#2100C2×C13711yescharacteristic, complemented
190#2094C2×C13711yescharacteristic
191#1271C2×C13711yescharacteristic, complemented
192#1271C2×C13711yescharacteristic, complemented
193#1270C2×C13711yescharacteristic
194C2×#126—21no—
195C2×#106—21no—
196C2×#102—21no—
197#399—21no—
198#396—21no—
199#394—21no—
200#390C22×C13711yescharacteristic
201#389—21no—
202#389C22×C13711yescharacteristic
203#389C22×C13711yescharacteristic
204#388—21no—
205#384C22×C13711yescharacteristic
206#340—21no—
207#340—21no—
208#340—21no—
209#340—21no—
210#336C22×C13711yescharacteristic
211#336C22×C13711yescharacteristic
212#336C22×C13711yescharacteristic
213C22×D4D4×C13711yescharacteristic
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217#126—42no—
218#126—42no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4D4×C13711yescharacteristic
227C2×C4⋊C4D4×C13711yescharacteristic
228C2×C4⋊C4D4×C13711yescharacteristic
229C2×C4⋊C4D4×C13711yescharacteristic
230C2×C4⋊C4D4×C13711yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C42—21no—
238C2×C42C23×C13711yescharacteristic
239#106—42no—
240#102—42no—
241#95—21no—
242#95—21no—
243#95—21no—
244C24—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—41no—
248C2×D4—41no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—41no—
255C22×C4C2×D4×C13711yescharacteristic
256C22×C4C2×D4×C13711yescharacteristic
257C22×C4C2×D4×C13711yescharacteristic
258C2×C8—22no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—81no—
283C4⋊C4SD16×C13712yes—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—81no—
291C22⋊C4—81no—
292C42—41no—
293C42—42no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23#120×C13711yescharacteristic
302D4—42no—
303D4—81no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4C2×SD16×C13711yescharacteristic
329C2×C4#135×C13711yescharacteristic
330C2×C4#135×C13711yescharacteristic
331C2×C4AGL(1,ℤ/8)×C13711yescharacteristic
332C2×C4#137×C13711yescharacteristic
333C2×C4#137×C13711yescharacteristic
334C8—42no—
335C8—81no—
336C8—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22#450×C13711yescharacteristic
345C22#452×C13711yescharacteristic
346C22#452×C13711yescharacteristic
347C22#453×C13711yescharacteristic
348C22#454×C13711yescharacteristic
349C22#454×C13711yescharacteristic
350C22#458×C13711yescharacteristic
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—42no—
358C4—81no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—161no—
363C2—161no—
364C2#1615×C13711yescharacteristic
365C2#1616×C13711yescharacteristic
366C2#1618×C13711yescharacteristic
367C2#1631×C13711yescharacteristic
368C2#1635×C13711yescharacteristic
369C2#1638×C13711yescharacteristic
370C2#1657×C13711yescharacteristic
subgroupG/H
center, socleC23×C137#120
commutator, Frattini, cosocleC2×C42C23×C137
prime radicalC22×D4×C137D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C137 ◁ C2×C42×C137 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #2100×C137C2semidirect
3. #2100×C137C2semidirect
5. #1271×C137C2semidirect
6. #1271×C137C2semidirect
185. #7828C137direct (N × H)
186. C137#7828direct (N × H)
188. #2100C2×C137semidirect
189. #2100C2×C137semidirect
191. #1271C2×C137semidirect
192. #1271C2×C137semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#2727×C1372
#2100×C1372
#2094×C1372
#1271×C1372
#1270×C1372
#7828137